Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 11E / c / Solution

The three jump vectors and their rates are $(1,0)$ at rate $\lambda$, $(-1,1)$ at rate $\gamma x_1$, and $(0,-1)$ at rate $\beta x_2$. Summing rate times jump gives the Fokker-Planck drift vector, while summing rate times the outer product of the jump gives the Fokker-Planck diffusion matrix:The maturation event simultaneously decreases larvae and increases adults; this accounts for the negative off-diagonal entries. Since $A$ is affine, its expectation is its value at the mean. Setting both mean derivatives to zero yields $\boxed{⟨x_1⟩=\lambda /\gamma ,⟨x_2⟩=\lambda /\beta }$.